### PDF handout - G. William Schwert

```GARCH Models
APS 425 - Advanced Managerial Data
Analysis
APS 425 – Fall 2014
GARCH Models
M d l
Instructor: G. William Schwert
585-275-2470
[email protected]
Autocorrelated Heteroskedasticity
• Suppose you have
h
regression
i residuals
id l
• Mean = 0, not autocorrelated
• Then, look at autocorrelations of the
absolute values of the residuals (or the
squares of the residuals)
• This tells you if there is heteroskedasticity
that varies over time
(c) Prof. G. William Schwert, 2002-2014
1
GARCH Models
APS 425 - Advanced Managerial Data
Analysis
Example: Xerox Stock Returns
Kurtosis
K
i and
d wide,
id then
h narrow, bands
b d iin plot
l are hints
hi off
conditional heteroskedasticity
XRX
.8
.6
160
Series: XRX
Sample 1961M07 2015M12
Observations 637
140
.4
120
100
80
60
40
20
.2
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.011243
0.008944
0.766487
-0.439834
0.102845
0.781145
9.576843
-.2
Jarque-Bera
Probability
1212.838
0.000000
-.4
.0
0
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
-.6
65
70
75
80
85
90
95
00
05
10
15
Example: Xerox Stock Returns
After estimating a regression with
just a constant for the Xerox
returns, the squared residuals
have small positive
autocorrelations
(c) Prof. G. William Schwert, 2002-2014
2
GARCH Models
APS 425 - Advanced Managerial Data
Analysis
GARCH Model
GARCH(1 1) which is not a bad starting point
Default model is GARCH(1,1),
GARCH Model
Conditional sd graph shows brief periods of
high volatility
.40
.35
.30
.25
.20
.15
.10
.05
65
70
75
80
85
90
95
00
05
10
Conditional standard deviation
(c) Prof. G. William Schwert, 2002-2014
3
GARCH Models
APS 425 - Advanced Managerial Data
Analysis
GARCH(1,1) Model
Rt =  + t
t ~ N(0, σt2)
σt2 = ω + α1 εt-12 + β1 σt-12
Where
is
i the
th mean off the
th returns
t
2
σt is the variance of the errors at time t
εt-12 is the squared error at time t-1
ω / (1 - β1 - α1) is the unconditional variance
α1 is the first (lag 1) ARCH parameter
β1 is the first (lag 1) GARCH parameter
GARCH(1,1) Model
This looks a lot like an ARMA(1,1) model for the squared
errors (as deviations from their forecasts),
νt = (t2 - σt2)
εt2 = ω + (α1 + β1) εt-12 + νt - β1 νt-1
Often the
Oft
th GARCH parameter
t β1 is
i close
l
to
t 1,
1 implying
i l i that
th t
the movements of the conditional variance away from its
long-run mean last a long time
For Xerox β1 = .75 and α1 = .18, so the implied AR(1)
parameter is about .93 and the MA(1) coefficient is .75
(c) Prof. G. William Schwert, 2002-2014
4
GARCH Models
APS 425 - Advanced Managerial Data
Analysis
GARCH Model Diagnostics
In Eviews,, most of the residual diagnostics
g
for GARCH models
are in terms of the standardized residuals [which should be
N(0,1)]
Note that kurtosis is smaller (still not 3, though)
80
Series: Standardized Residuals
Sample 1961M08 2014M08
Observations 637
70
60
50
40
30
20
10
Mean
M
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.038129
0 038129
-0.033813
3.240825
-4.201253
1.000207
-0.080484
3.987784
Jarque-Bera
Probability
26.58485
0.000002
0
-4
-3
-2
-1
0
1
2
3
GARCH Model Diagnostics
g
The correlogram
for the standardized
squared residuals now looks
better
(c) Prof. G. William Schwert, 2002-2014
5
GARCH Models
APS 425 - Advanced Managerial Data
Analysis
EGARCH(1,1) Model
This model basicallyy models the logg
of the variance (or standard
deviation) as a function of the
lagged log(variance/std dev) and
the lagged absolute error from the
regression model
It also allows the response to the
lagged
l
d error to
t be
b asymmetric,
t i so
that positive regression residuals
can have a different effect on
variance than an equivalent
negative residual
EGARCH(1,1) Model
“GARCH” is the variance the
residuals at time t
The persistence parameter, c(5),
is very large, implying that the
variance moves slowly
through time
Th asymmetry
t coefficient,
ffi i t c(4),
(4)
The
is negative, implying that the
variance goes up more after
negative residuals (stock
returns) than after positive
residuals (returns)
(c) Prof. G. William Schwert, 2002-2014
6
GARCH Models
APS 425 - Advanced Managerial Data
Analysis
EGARCH Model Diagnostics
The correlogram for the standardized
id l still
till looks
l k pretty
tt goodd
squaredd residuals
.30
.25
.20
.15
.10
.05
.00
65
70
75
80
85
90
95
00
05
10
Conditional standard deviation
EGARCH Model Diagnostics
In Eviews, most of the residual diagnostics for GARCH models are in terms of the
standardized
t d di d residuals
id l [which
[ hi h should
h ld be
b N(0,1)]
N(0 1)]
Note that kurtosis is smaller (still not 3, though)
80
Series: Standardized Residuals
Sample 1961M08 2014M08
Observations 637
70
60
50
40
30
20
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.017141
0.007866
3.205491
-4.443921
1.000690
-0.062749
3.813891
Jarque-Bera
Probability
17.99972
0.000123
0
-4
-3
-2
(c) Prof. G. William Schwert, 2002-2014
-1
0
1
2
3
7
GARCH Models
APS 425 - Advanced Managerial Data
Analysis
EGARCH Model Extensions
Plotting the log of Xerox’s stock price on the right axis, versus the two estimates of
th conditional
the
diti l standard
t d d deviation
d i ti [from
[f
GARCH(1,1)
GARCH(1 1) andd EGARCH(1,1)],
EGARCH(1 1)] you
can see that the crash in the stock price occurs at the same time as the spike in
volatility, and volatility declined as the stock price slowly recovered
.4
.3
.2
5
.1
1
4
.0
3
2
1
0
65
70
75
80
SD01
85
90
SD02
95
00
05
10
15
LOG(XRXP)
EGARCH Model Extensions
Include the lagged log of Xerox’s stock price as an additional variable in the
EGARCH equation,
ti bbutt it ddoesn’t
dd much
h
(c) Prof. G. William Schwert, 2002-2014
8
GARCH Models
APS 425 - Advanced Managerial Data
Analysis